QuaRK: A Quantum Reservoir Kernel for Time Series Learning

TL;DR

QuaRK, an end-to-end framework that couples a hardware-realistic quantum reservoir featurizer with a kernel-based readout scheme, provides learning-theoretic generalization guarantees for dependent temporal data, linking design and resource choices to finite-sample performance, thereby offering principled guidance for building reliable temporal learners.

Abstract

Quantum reservoir computing offers a promising route for time series learning by modelling sequential data via rich quantum dynamics while the only training required happens at the level of a lightweight classical readout. However, studies featuring efficient and implementable quantum reservoir architectures along with model learning guarantees remain scarce in the literature. To close this gap, we introduce QuaRK, an end-to-end framework that couples a hardware-realistic quantum reservoir featurizer with a kernel-based readout scheme. Given a sequence of sample points, the reservoir injects the points one after the other to yield a compact feature vector from efficiently measured k-local observables using classical shadow tomography, after which a classical kernel-based readout learns the target mapping with explicit regularization and fast optimization. The resulting pipeline exposes clear computational knobs -- circuit width and depth as well as the measurement budget -- while preserving the flexibility of kernel methods to model nonlinear temporal functionals and being scalable to high-dimensional data. We further provide learning-theoretic generalization guarantees for dependent temporal data, linking design and resource choices to finite-sample performance, thereby offering principled guidance for building reliable temporal learners. Empirical experiments validate QuaRK and illustrate the predicted interpolation and generalization behaviours on synthetic beta-mixing time series tasks.

Figures (5)

• Figure 1: Circuit for the unitary evolution block $V(x)$ on three qubits with ring connectivity. First, classical features are encoded via single-qubit angle encoding $R_y(\theta(z_j))$ on each $q_j$. Then the Ising-like unitary $W$ ( \ref{['eq:ising-unitary']}) is shown as ZZ couplings on the edges $(1,2)$, $(2,3)$, and $(3,1)$, followed by local rotations $R_z(\vartheta_z)$ and $R_x(\vartheta_x)$.
• Figure 2: SWAP-dilation realization of the contractive channel $E_\lambda$ acting on a 3-qubit reservoir state $\rho$ (top wires). The ancilla register (middle wires) is reset to $\ket{0}$ and prepared in $\ket{+}^{\otimes 3}$ via Hadamards. A coin qubit (bottom wire) is reset and rotated by $R_y(\theta_\lambda)$, where $\theta_\lambda = 2\arcsin\!\sqrt{1-\lambda}$ so that $\Pr[c=1]=1-\lambda$ and $\Pr[c=0]=\lambda$. Conditioned on $c=1$, controlled-SWAPs exchange each reservoir qubit $q_i$ with its corresponding ancilla qubit $a_i$.
• Figure 3: Training MSE versus the readout regularization $\lambda_{\rm reg}$ for the three functionals, featuring a sharp transition into the interpolation regime around $\lambda_{\rm reg} \approx 10^{-1}$, where the curves reach numerical zero.
• Figure 4: Predicted vs true labels on a subset of training windows for the three functionals from the interpolation regime. Numerical errors reach values of ${\rm MSE} \in [10^{-14}, 10^{-12}]$ confirming perfect task learning.
• Figure 5: Test MSE vs training-set size $N$ reported for the three tasks, obtained by sweeping $N$ while keeping the other experimental choices, and reported for a same held-out test split.

Theorems & Definitions (9)

• Theorem 1: Effective learning
• Theorem 2: Generalization on weakly-dependent data
• Claim 1: Closed form reservoir recursion
• Proposition 1: Set-injectivity
• Claim 2: $\beta$-mixing property of the windows process
• Theorem 3: Mohri–Rostamizadeh mohri2008rademacher
• Corollary 1: Non-i.i.d. generalization for MSE and RKHS-ball readouts
• Proposition 2: Geometric decay of dependence of reservoir outputs on inputs
• Corollary 2: Deviation of the window risk from the statistical one